3.112 \(\int \frac {d+e x+f x^2}{(g+h x) (a+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}-\frac {\left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{\left (a h^2+c g^2\right )^{3/2}} \]

[Out]

-(d*h^2-e*g*h+f*g^2)*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/(a*h^2+c*g^2)^(3/2)+(-a*(a*f*h-
c*d*h+c*e*g)+c*(a*e*h-a*f*g+c*d*g)*x)/a/c/(a*h^2+c*g^2)/(c*x^2+a)^(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1647, 12, 725, 206} \[ -\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}-\frac {\left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{\left (a h^2+c g^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2)/((g + h*x)*(a + c*x^2)^(3/2)),x]

[Out]

-((a*(c*e*g - c*d*h + a*f*h) - c*(c*d*g - a*f*g + a*e*h)*x)/(a*c*(c*g^2 + a*h^2)*Sqrt[a + c*x^2])) - ((f*g^2 -
 e*g*h + d*h^2)*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(c*g^2 + a*h^2)^(3/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
 (a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
 e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
 (c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}+\frac {\int \frac {a c \left (f g^2-e g h+d h^2\right )}{\left (c g^2+a h^2\right ) (g+h x) \sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}+\frac {\left (f g^2-e g h+d h^2\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{c g^2+a h^2}\\ &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}-\frac {\left (f g^2-e g h+d h^2\right ) \operatorname {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{c g^2+a h^2}\\ &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}-\frac {\left (f g^2-e g h+d h^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{\left (c g^2+a h^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 137, normalized size = 0.99 \[ \frac {a^2 (-f) h+a c (d h-e g+e h x-f g x)+c^2 d g x}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}-\frac {\left (h (d h-e g)+f g^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{\left (a h^2+c g^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2)/((g + h*x)*(a + c*x^2)^(3/2)),x]

[Out]

(-(a^2*f*h) + c^2*d*g*x + a*c*(-(e*g) + d*h - f*g*x + e*h*x))/(a*c*(c*g^2 + a*h^2)*Sqrt[a + c*x^2]) - ((f*g^2
+ h*(-(e*g) + d*h))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(c*g^2 + a*h^2)^(3/2)

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fricas [B]  time = 4.00, size = 721, normalized size = 5.22 \[ \left [\frac {{\left (a^{2} c f g^{2} - a^{2} c e g h + a^{2} c d h^{2} + {\left (a c^{2} f g^{2} - a c^{2} e g h + a c^{2} d h^{2}\right )} x^{2}\right )} \sqrt {c g^{2} + a h^{2}} \log \left (\frac {2 \, a c g h x - a c g^{2} - 2 \, a^{2} h^{2} - {\left (2 \, c^{2} g^{2} + a c h^{2}\right )} x^{2} - 2 \, \sqrt {c g^{2} + a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{h^{2} x^{2} + 2 \, g h x + g^{2}}\right ) - 2 \, {\left (a c^{2} e g^{3} + a^{2} c e g h^{2} - {\left (a c^{2} d - a^{2} c f\right )} g^{2} h - {\left (a^{2} c d - a^{3} f\right )} h^{3} - {\left (a c^{2} e g^{2} h + a^{2} c e h^{3} + {\left (c^{3} d - a c^{2} f\right )} g^{3} + {\left (a c^{2} d - a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} g^{4} + 2 \, a^{3} c^{2} g^{2} h^{2} + a^{4} c h^{4} + {\left (a c^{4} g^{4} + 2 \, a^{2} c^{3} g^{2} h^{2} + a^{3} c^{2} h^{4}\right )} x^{2}\right )}}, -\frac {{\left (a^{2} c f g^{2} - a^{2} c e g h + a^{2} c d h^{2} + {\left (a c^{2} f g^{2} - a c^{2} e g h + a c^{2} d h^{2}\right )} x^{2}\right )} \sqrt {-c g^{2} - a h^{2}} \arctan \left (\frac {\sqrt {-c g^{2} - a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{a c g^{2} + a^{2} h^{2} + {\left (c^{2} g^{2} + a c h^{2}\right )} x^{2}}\right ) + {\left (a c^{2} e g^{3} + a^{2} c e g h^{2} - {\left (a c^{2} d - a^{2} c f\right )} g^{2} h - {\left (a^{2} c d - a^{3} f\right )} h^{3} - {\left (a c^{2} e g^{2} h + a^{2} c e h^{3} + {\left (c^{3} d - a c^{2} f\right )} g^{3} + {\left (a c^{2} d - a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{a^{2} c^{3} g^{4} + 2 \, a^{3} c^{2} g^{2} h^{2} + a^{4} c h^{4} + {\left (a c^{4} g^{4} + 2 \, a^{2} c^{3} g^{2} h^{2} + a^{3} c^{2} h^{4}\right )} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)/(h*x+g)/(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[1/2*((a^2*c*f*g^2 - a^2*c*e*g*h + a^2*c*d*h^2 + (a*c^2*f*g^2 - a*c^2*e*g*h + a*c^2*d*h^2)*x^2)*sqrt(c*g^2 + a
*h^2)*log((2*a*c*g*h*x - a*c*g^2 - 2*a^2*h^2 - (2*c^2*g^2 + a*c*h^2)*x^2 - 2*sqrt(c*g^2 + a*h^2)*(c*g*x - a*h)
*sqrt(c*x^2 + a))/(h^2*x^2 + 2*g*h*x + g^2)) - 2*(a*c^2*e*g^3 + a^2*c*e*g*h^2 - (a*c^2*d - a^2*c*f)*g^2*h - (a
^2*c*d - a^3*f)*h^3 - (a*c^2*e*g^2*h + a^2*c*e*h^3 + (c^3*d - a*c^2*f)*g^3 + (a*c^2*d - a^2*c*f)*g*h^2)*x)*sqr
t(c*x^2 + a))/(a^2*c^3*g^4 + 2*a^3*c^2*g^2*h^2 + a^4*c*h^4 + (a*c^4*g^4 + 2*a^2*c^3*g^2*h^2 + a^3*c^2*h^4)*x^2
), -((a^2*c*f*g^2 - a^2*c*e*g*h + a^2*c*d*h^2 + (a*c^2*f*g^2 - a*c^2*e*g*h + a*c^2*d*h^2)*x^2)*sqrt(-c*g^2 - a
*h^2)*arctan(sqrt(-c*g^2 - a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a)/(a*c*g^2 + a^2*h^2 + (c^2*g^2 + a*c*h^2)*x^2))
 + (a*c^2*e*g^3 + a^2*c*e*g*h^2 - (a*c^2*d - a^2*c*f)*g^2*h - (a^2*c*d - a^3*f)*h^3 - (a*c^2*e*g^2*h + a^2*c*e
*h^3 + (c^3*d - a*c^2*f)*g^3 + (a*c^2*d - a^2*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/(a^2*c^3*g^4 + 2*a^3*c^2*g^2*h^2
 + a^4*c*h^4 + (a*c^4*g^4 + 2*a^2*c^3*g^2*h^2 + a^3*c^2*h^4)*x^2)]

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giac [B]  time = 0.29, size = 294, normalized size = 2.13 \[ \frac {\frac {{\left (c^{3} d g^{3} - a c^{2} f g^{3} + a c^{2} d g h^{2} - a^{2} c f g h^{2} + a c^{2} g^{2} h e + a^{2} c h^{3} e\right )} x}{a c^{3} g^{4} + 2 \, a^{2} c^{2} g^{2} h^{2} + a^{3} c h^{4}} + \frac {a c^{2} d g^{2} h - a^{2} c f g^{2} h + a^{2} c d h^{3} - a^{3} f h^{3} - a c^{2} g^{3} e - a^{2} c g h^{2} e}{a c^{3} g^{4} + 2 \, a^{2} c^{2} g^{2} h^{2} + a^{3} c h^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, {\left (f g^{2} + d h^{2} - g h e\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} h + \sqrt {c} g}{\sqrt {-c g^{2} - a h^{2}}}\right )}{{\left (c g^{2} + a h^{2}\right )} \sqrt {-c g^{2} - a h^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)/(h*x+g)/(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

((c^3*d*g^3 - a*c^2*f*g^3 + a*c^2*d*g*h^2 - a^2*c*f*g*h^2 + a*c^2*g^2*h*e + a^2*c*h^3*e)*x/(a*c^3*g^4 + 2*a^2*
c^2*g^2*h^2 + a^3*c*h^4) + (a*c^2*d*g^2*h - a^2*c*f*g^2*h + a^2*c*d*h^3 - a^3*f*h^3 - a*c^2*g^3*e - a^2*c*g*h^
2*e)/(a*c^3*g^4 + 2*a^2*c^2*g^2*h^2 + a^3*c*h^4))/sqrt(c*x^2 + a) - 2*(f*g^2 + d*h^2 - g*h*e)*arctan(((sqrt(c)
*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c*g^2 + a*h^2)*sqrt(-c*g^2 - a*h^2))

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maple [B]  time = 0.02, size = 862, normalized size = 6.25 \[ \frac {c d g x}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, a}-\frac {c e \,g^{2} x}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, a h}+\frac {c f \,g^{3} x}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, a \,h^{2}}-\frac {d h \ln \left (\frac {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {e g \ln \left (\frac {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {f \,g^{2} \ln \left (\frac {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, h}+\frac {d h}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {e g}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {f \,g^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, h}+\frac {e x}{\sqrt {c \,x^{2}+a}\, a h}-\frac {f g x}{\sqrt {c \,x^{2}+a}\, a \,h^{2}}-\frac {f}{\sqrt {c \,x^{2}+a}\, c h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e*x+d)/(h*x+g)/(c*x^2+a)^(3/2),x)

[Out]

-1/h*f/c/(c*x^2+a)^(1/2)+1/h*e*x/a/(c*x^2+a)^(1/2)-1/h^2*f*g*x/a/(c*x^2+a)^(1/2)+h/(a*h^2+c*g^2)/(-2*(x+g/h)*c
*g/h+(x+g/h)^2*c+(a*h^2+c*g^2)/h^2)^(1/2)*d-1/(a*h^2+c*g^2)/(-2*(x+g/h)*c*g/h+(x+g/h)^2*c+(a*h^2+c*g^2)/h^2)^(
1/2)*e*g+1/h/(a*h^2+c*g^2)/(-2*(x+g/h)*c*g/h+(x+g/h)^2*c+(a*h^2+c*g^2)/h^2)^(1/2)*f*g^2+g/(a*h^2+c*g^2)/a/(-2*
(x+g/h)*c*g/h+(x+g/h)^2*c+(a*h^2+c*g^2)/h^2)^(1/2)*x*c*d-1/h*g^2/(a*h^2+c*g^2)/a/(-2*(x+g/h)*c*g/h+(x+g/h)^2*c
+(a*h^2+c*g^2)/h^2)^(1/2)*x*c*e+1/h^2*g^3/(a*h^2+c*g^2)/a/(-2*(x+g/h)*c*g/h+(x+g/h)^2*c+(a*h^2+c*g^2)/h^2)^(1/
2)*x*c*f-h/(a*h^2+c*g^2)/((a*h^2+c*g^2)/h^2)^(1/2)*ln((-2*(x+g/h)*c*g/h+2*(a*h^2+c*g^2)/h^2+2*((a*h^2+c*g^2)/h
^2)^(1/2)*(-2*(x+g/h)*c*g/h+(x+g/h)^2*c+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*d+1/(a*h^2+c*g^2)/((a*h^2+c*g^2)/h^
2)^(1/2)*ln((-2*(x+g/h)*c*g/h+2*(a*h^2+c*g^2)/h^2+2*((a*h^2+c*g^2)/h^2)^(1/2)*(-2*(x+g/h)*c*g/h+(x+g/h)^2*c+(a
*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*e*g-1/h/(a*h^2+c*g^2)/((a*h^2+c*g^2)/h^2)^(1/2)*ln((-2*(x+g/h)*c*g/h+2*(a*h^2
+c*g^2)/h^2+2*((a*h^2+c*g^2)/h^2)^(1/2)*(-2*(x+g/h)*c*g/h+(x+g/h)^2*c+(a*h^2+c*g^2)/h^2)^(1/2))/(x+g/h))*f*g^2

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maxima [B]  time = 0.62, size = 453, normalized size = 3.28 \[ \frac {c f g^{3} x}{\sqrt {c x^{2} + a} a c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{2} h^{4}} - \frac {c e g^{2} x}{\sqrt {c x^{2} + a} a c g^{2} h + \sqrt {c x^{2} + a} a^{2} h^{3}} + \frac {c d g x}{\sqrt {c x^{2} + a} a c g^{2} + \sqrt {c x^{2} + a} a^{2} h^{2}} + \frac {f g^{2}}{\sqrt {c x^{2} + a} c g^{2} h + \sqrt {c x^{2} + a} a h^{3}} - \frac {e g}{\sqrt {c x^{2} + a} c g^{2} + \sqrt {c x^{2} + a} a h^{2}} + \frac {d}{\frac {\sqrt {c x^{2} + a} c g^{2}}{h} + \sqrt {c x^{2} + a} a h} - \frac {f g x}{\sqrt {c x^{2} + a} a h^{2}} + \frac {e x}{\sqrt {c x^{2} + a} a h} + \frac {f g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} - \frac {e g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{2}} + \frac {d \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h} - \frac {f}{\sqrt {c x^{2} + a} c h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)/(h*x+g)/(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

c*f*g^3*x/(sqrt(c*x^2 + a)*a*c*g^2*h^2 + sqrt(c*x^2 + a)*a^2*h^4) - c*e*g^2*x/(sqrt(c*x^2 + a)*a*c*g^2*h + sqr
t(c*x^2 + a)*a^2*h^3) + c*d*g*x/(sqrt(c*x^2 + a)*a*c*g^2 + sqrt(c*x^2 + a)*a^2*h^2) + f*g^2/(sqrt(c*x^2 + a)*c
*g^2*h + sqrt(c*x^2 + a)*a*h^3) - e*g/(sqrt(c*x^2 + a)*c*g^2 + sqrt(c*x^2 + a)*a*h^2) + d/(sqrt(c*x^2 + a)*c*g
^2/h + sqrt(c*x^2 + a)*a*h) - f*g*x/(sqrt(c*x^2 + a)*a*h^2) + e*x/(sqrt(c*x^2 + a)*a*h) + f*g^2*arcsinh(c*g*x/
(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^3) - e*g*arcsinh(c*g*x/(sqrt
(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^2) + d*arcsinh(c*g*x/(sqrt(a*c)*a
bs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h) - f/(sqrt(c*x^2 + a)*c*h)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f\,x^2+e\,x+d}{\left (g+h\,x\right )\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x + f*x^2)/((g + h*x)*(a + c*x^2)^(3/2)),x)

[Out]

int((d + e*x + f*x^2)/((g + h*x)*(a + c*x^2)^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x + f x^{2}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (g + h x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)/(h*x+g)/(c*x**2+a)**(3/2),x)

[Out]

Integral((d + e*x + f*x**2)/((a + c*x**2)**(3/2)*(g + h*x)), x)

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